1.4 MAGNETIC INTERACTIONS
39
1.4 Magnetic interactions
In the metallic state, the 4f electrons on a rare earth ion are subjected
to a variety of interactions with their surroundings. These forces may
be broadly classified into two categories. The single-ion interactions act
independently at each ionic site, so that their influence on the state of
the 4f electrons at a particular site is unaffected by the magnetic state
of its neighbours. The corresponding contribution to the Hamiltonian
therefore contains sums over terms located at the ionic sites i of the
crystal, but without any coupling between different ions. On the other
hand, the two-ion interactions couple the 4f -electron clouds at pairs of
ions, giving terms which involve two sites i and j.
The charge distribution around an ion produces an electric field,
with the local point-symmetry, which acts on the 4f electrons and gives
rise to the large magnetic anisotropies which are characteristic of the rare
earth metals. This crystal field makes a contribution to the potential
energy of a 4f electron with charge −e
eρ(R)
dR,
(1.4.1)
vcf (r) = −
|r − R|
where ρ(R) is the charge density of the surrounding electrons and nuclei.
If these do not penetrate the 4f charge cloud, vcf (r) is a solution of
Laplace’s equation, and may be expanded in spherical harmonics as
vcf (r) =
l
Am
l r Ylm (r̂),
(1.4.2)
lm
where
Am
l
4π
= −(−1)
2l + 1
m
eρ(R)
Y
(R̂) dR,
Rl+1 l−m
(1.4.3)
which is a special case of the multipole expansion (1.3.7). We can
thus look upon (1.4.2) as arising from the interaction of the multipoles
rl Ylm (r̂) of the 4f electrons with the appropriate components of the
electric field. If part of the charge which is responsible for the crystal
field lies within the 4f cloud, vcf (r) can still be expanded in spherical
40
1. ELEMENTS OF RARE EARTH MAGNETISM
harmonics with the appropriate symmetry, but the coefficients are not
generally proportional to rl , nor to (1.4.3).
As the crystal-field energy is small compared to the spin–orbit splitting, its effects on the eigenstates of the system are adequately accounted
for by first-order perturbation theory. Since f electrons cannot have
multipole distributions with l > 6, the properties of the spherical harmonics ensure that the corresponding matrix elements of (1.4.2) vanish.
Even so, the calculation of those that remain from the electronic wavefunctions would be a formidable task, even if the surrounding charge
distribution were known, if the ubiquitous Wigner–Eckart theorem did
not once again come to the rescue. As first pointed out by Stevens
(1952), provided that we remain within a manifold of constant J, in
this case the ground-state multiplet, the matrix elements of vcf (r) are
proportional to those of operator equivalents, written in terms of the J
operators. We may thus replace (1.4.2) by
1/2
2l + 1
l
lm (Ji ),
O
Am
α
r
(1.4.4)
Hcf =
l
l
4π
i
lm
where we have also summed over the ions. The Stevens factors αl depend on the form of the electronic charge cloud through L, S and J, and
on l, but not on m. They are frequently denoted α, β, and γ when l is 2,
4, and 6 respectively, and their values for the magnetic rare earth ions
are given in Table 1.4. The expectation value rl is an average over the
(J) are obtained from the spherical
4f states. The Racah operators O
lm
harmonics, multiplied by (4π/2l + 1)1/2 , by writing them in terms of
Table 1.4. Stevens factors for rare earth ions.
Ion+++
Ce
Pr
Nd
Pm
Sm
Tb
Dy
Ho
Er
Tm
Yb
α ×102
β ×104
−5.714
−2.101
−0.6428
0.7714
4.127
−1.0101
−0.6349
−0.2222
0.2540
1.0101
3.175
63.49
−7.346
−2.911
4.076
25.01
1.224
−0.5920
−0.3330
0.4440
1.632
−17.32
γ ×106
0
60.99
−37.99
60.78
0
−1.121
1.035
−1.294
2.070
−5.606
148.0
1.4 MAGNETIC INTERACTIONS
41
Cartesian coordinates and replacing (x, y, z) by (Jx , Jy , Jz ), with an appropriate symmetrization to take account of the non-commutation of the
J operators. They have been tabulated for l-values up to 8 by Lindgård
and Danielsen (1974).
Following the customary practice, we shall generally use not the
Racah operators, which are tensor operators transforming under rotations like spherical harmonics, but the Stevens operators Olm (J), which
transform like the real tesseral harmonics Tlm . If we define corresponding operators for m zero or positive as:
l0
Tl0 = O
1 lm
Ol−m + (−1)m O
2
i ,
= √ Ol−m − (−1)m O
lm
2
c
Tlm
= √
s
Tlm
(1.4.5)
the Stevens operators for positive and negative m are proportional rec
s
spectively to Tlm
and Tl|m|
. There is some ambiguity in the literature
about the proportionality constants, but we have used the standard definitions of the Stevens operators in Table 1.5, see also Hutchings (1964).
In terms of these operators, we may write the crystal-field Hamiltonian
Hcf =
i
Blm Olm (Ji ).
(1.4.6a)
lm
The crystal-field parameters Blm can in principle be calculated from the
charge distribution in the metal, but in practice attempts to do so have
met with limited success. The difficulties are two-fold. The charge density on the surroundings of an ion is not easy to determine with the
necessary accuracy, and the approximations normally used in the calculation of the electronic structure of a metal, in particular the assumption
that the charge distribution in the atomic polyhedron is spherically symmetric, are inadequate for the purpose. Furthermore, a redistribution
of the charge within the cell can modify the electric fields experienced
by the 4f electrons, and such shielding effects are again very difficult
to estimate. It is therefore necessary to appeal to relatively crude models, such as the instructive but quite unjustified point-charge model, in
which an adjustable charge is placed on each lattice site, or alternatively
to regard the Blm as parameters to be determined from experiment.
Fortunately, the number of such parameters is strongly restricted by
symmetry. We shall be concerned almost exclusively with the hexagonal
structures of Fig. 1.3, and in defining the Stevens operators, we have
used a Cartesian system in which the (x, y, z)-directions are along the
1. ELEMENTS OF RARE EARTH MAGNETISM
42
Table 1.5. Stevens operators. X ≡ J(J + 1) and J± ≡ Jx ± iJy .
O21
1 2
2
(J + J−
)
2 +
1
= (Jz Jx + Jx Jz )
2
O20
= 3Jz2 − X
O22
=
1
(Jz Jy + Jy Jz )
2
1 2
2
=
(J − J−
)
2i +
1 4
4
= (J+
+ J−
)
2
1
2
2
2
2
(7Jz2 − X − 5)(J+
=
+ J−
) + (J+
+ J−
)(7Jz2 − X − 5)
4
O2−1 =
O2−2
O44
O42
O40
O4−2
O4−4
= 35Jz4 − (30X − 25)Jz2 + 3X 2 − 6X
1
2
2
2
2
(7Jz2 − X − 5)(J+
=
− J−
) + (J+
− J−
)(7Jz2 − X − 5)
4i
1 4
4
=
(J − J−
)
2i +
O60
= 231Jz6 − (315X − 735)Jz4 + (105X 2 − 525X + 294)Jz2
−5X 3 + 40X 2 − 60X
O66
=
1 6
6
(J + J−
)
2 +
crystallographic (a, b, c)-axes specified in the previous section. However, it will later be convenient to rotate the z-axis into the magnetization direction, and instead orient the crystallographic (a, b, c)-axes
along the (ξ, η, ζ)-Cartesian directions. For an ion with hexagonal pointsymmetry, as in the hcp structure or on the hexagonal sites of the dhcp
structure, the crystal field is specified by 4 parameters:
Bl0 Ol0 (Ji ) + B66 O66 (Ji ) .
(1.4.6b)
Hcf =
i
l=2,4,6
The Hamiltonian (1.4.6) lifts the degeneracy of the ionic |JMJ > states
and, since it is expressed in terms of J operators, whose matrix elements
between these states may be determined by straightforward calculation,
it may readily be diagonalized to yield the crystal-field energies and
eigenfunctions. The Blm may then be used as adjustable parameters to
reproduce the available experimental information on these eigenstates.
As an example, we show in Fig. 1.16 the splitting of the nine |4MJ >
states in Pr by the crystal fields acting on the hexagonal sites. This level
scheme was derived from values of the crystal-field parameters adjusted
1.4 MAGNETIC INTERACTIONS
43
Fig. 1.16. The crystal-field splitting of the nine |4MJ > states on
the hexagonal sites in dhcp Pr. The
wavefunctions are written in terms
of a basis |MJ > corresponding to
quantization along the c-direction.
to account for a variety of experimental phenomena (Houmann et al.
1979).
If the lattice is strained, the crystal fields, and indeed all the other
magnetic interactions which we shall discuss, are modified. In consequence, there is a magnetoelastic coupling between the moments and
the strain, which can have profound consequences for rare earth magnetism. Magnetoelastic effects are manifested in both single-ion and
two-ion terms in the Hamiltonian, though we shall mostly be concerned
with the former. The elastic energy is quadratic in the strain, measured
relative to the equilibrium configuration in the absence of magnetic interactions. The magnetoelastic energy is linear in the strain and the
competition between the two effects may lead to some equilibrium strain
or magnetostriction. Because of their moderate elastic constants and the
large orbital component in their moments, the lanthanide metals display
the largest known magnetostrictions.
Following Callen and Callen (1965), it is convenient to develop the
theory in terms of the irreducible strains for hexagonal point-symmetry,
1. ELEMENTS OF RARE EARTH MAGNETISM
44
which are related to the Cartesian strains as follows:
α1 = 11 + 22 + 33
α2 = 13 (233 − 11 − 22 )
γ1 = 12 (11 − 22 )
(1.4.7)
γ2 = 12
ε1 = 13
ε2 = 23 ,
where we have adopted the conventional notation of designating the
Cartesian axes (ξ, η, ζ) by (1, 2, 3). The α-strains are thus symmetryconserving dilatations, the γ-strains distort the hexagonal symmetry of
the basal plane, and the ε-strains shear the c-axis. The elastic energy
may then be written
Hel = N 12 cα1 2α1 + cα3 α1 α2 + 12 cα2 2α2
(1.4.8)
+ 12 cγ (2γ1 + 2γ2 ) + 12 cε (2ε1 + 2ε2 ) ,
where we have defined irreducible elastic stiffness constants per ion, related to the five independent Cartesian constants by
cα1 = 19 (2c11 + 2c12 + 4c13 + c33 )V /N
cα2 = 12 (c11 + c12 − 4c13 + 2c33 )V /N
cα3 = 13 (−c11 − c12 + c13 + c33 )V /N
cγ = 2(c11 − c12 )V /N
(1.4.9)
cε = 4c44 V /N.
The contributions to the single-ion magnetoelastic Hamiltonian,
corresponding to the different irreducible strains, are
α
l
l
Hme
=−
α1 + Bα2
α2 Ol0 (Ji )
Bα1
i
γ
Hme
l=2,4,6
66
66
α1 + Bα2
α2 O66 (Ji )
+ Bα1
2
l
Ol (Ji )γ1 + Ol−2 (Ji )γ2
=−
Bγ2
i
(1.4.10)
l=2,4,6
+
4
l
Ol (Ji )γ1 − Ol−4 (Ji )γ2
(1.4.11)
Bγ4
l=4,6
ε
Hme
=−
i
1
l
Bε1
Ol (Ji )ε1 + Ol−1 (Ji )ε2
l=2,4,6
+Bε5 O65 (Ji )ε1 − O6−5 (Ji )ε2 . (1.4.12)
1.4 MAGNETIC INTERACTIONS
45
The operators in the α-strain term are the same as those in the crystalfield Hamiltonian (1.4.6b), and the associated magnetoelastic effects may
thus be considered as a strain-dependent renormalization of the crystalfield parameters, except that these interactions may mediate a dynamical
coupling between the magnetic excitations and the phonons. The other
two terms may have the same effect, but they also modify the symmetry and, as we shall see, can therefore qualitatively influence both the
magnetic structures and excitations.
It is the two-ion couplings which are primarily responsible for cooperative effects and magnetic ordering in the rare earths, and of these
the most important is the indirect exchange, by which the moments
on pairs of ions are coupled through the intermediary of the conduction
electrons. The form of this coupling can be calculated straightforwardly,
provided that we generalize (1.3.22) slightly to
2
I(r − Ri ) Si · s(r)dr = − Hi (r) · µ(r)dr, (1.4.13)
Hsf (i) = −
N
where N is the number of ions, s(r) is the conduction-electron spin
density, and the exchange integral I(r − Ri ) is determined by the overlap of the 4f and conduction-electron charge clouds. This expression,
whose justification and limitations will be discussed in Section 5.7, can
be viewed as arising from the action of the effective inhomogeneous magnetic field
1
I(r − Ri )Si
Hi (r) =
N µB
on the conduction-electron moment density µ(r) = 2µB s(r). The spin
at Ri generates a moment at r, whose Cartesian components are given
by
1 (1.4.14)
χαβ (r − r )Hiβ (r )dr ,
µiα (r) =
V
β
where χ is the nonlocal susceptibility tensor for the conduction electrons
and V the volume. This induced moment interacts through Hsf (j) with
the spin Sj , leading to a coupling
1 (1.4.15)
Hjα (r)χαβ (r − r )Hiβ (r )drdr .
H(ij) = −
V
αβ
If we neglect, for the moment, the spin–orbit coupling of the conduction
electrons, and the crystal is unmagnetized, χαβ becomes a scalar. We
define the Fourier transforms:
1
χ(q) =
χ(r) e−iq·r dr
V
(1.4.16)
V
iq·r
χ(r) =
dq
χ(q)
e
(2π)3
1. ELEMENTS OF RARE EARTH MAGNETISM
46
and
1
I(q) =
V
I(r) e−iq·r dr.
(1.4.17)
Summing over the lattice sites, counting each interaction once only, we
find that the indirect-exchange interaction takes the familiar isotropic
Heisenberg form:
V 1 V
χ(q)I(q)I(−q) eiq·(Ri −Rj ) Si ·Sj dq
Hff = −
2 N 2 µ2B (2π)3 ij
=−
1 JS (q) eiq·(Ri −Rj ) Si ·Sj
2N
q
=−
1
2
ij
JS (ij) Si ·Sj ,
(1.4.18)
ij
where
JS (ij) =
1 JS (q) eiq·(Ri −Rj )
N
(1.4.19)
V
|I(q)|2 χ(q).
N µ2B
(1.4.20)
and
JS (q) =
q
In the presence of an orbital moment, it is convenient to express
(1.4.18) in terms of J rather than S, which we may do within the groundstate multiplet by using (1.2.29) to project S on to J, obtaining
Hff = −
1
J (ij) Ji ·Jj ,
2
(1.4.21)
ij
with
1 JS (q ) ,
J (q) = (g − 1)2 JS (q) −
N
(1.4.22)
q
where we have also subtracted the interaction of the ith moment with
itself, as this term only leads to the constant contribution to the Hamiltonian; − 12 (g − 1)2 N JS (ii)J(J + 1). The origin of the indirect exchange in the polarization of the conduction-electron gas by the spin
on one ion, and the influence of this polarization on the spin of a
second ion, is apparent in the expression (1.4.20) for JS (q). As we
shall see, it is the Fourier transform [J (q) − J (0)] which may be directly deduced from measurements of the dispersion relations for the
magnetic excitations, and its experimentally determined variation with
q in the c-direction for the heavy rare earths is shown in Fig. 1.17.
1.4 MAGNETIC INTERACTIONS
47
Fig. 1.17. The exchange interaction JS (q) − JS (0), determined experimentally in the magnetic heavy rare earth metals. The magnitude
of the peak, which stabilizes the observed periodic magnetic structures,
increases monotonically with atomic number.
A notable feature is the maximum which, except in Gd, occurs at nonzero q and, as discussed in the following section, is responsible for stabilizing the periodic magnetic structures in the metals. In the approximation which we have used, the conduction-electron susceptibility is given
by
χ(q) =
fnk − fn k−q
2µ2B ,
V
εn (k − q) − εn (k)
nn k
(1.4.23)
48
1. ELEMENTS OF RARE EARTH MAGNETISM
where fnk is the Fermi–Dirac function. It is clear that a large contribution to the sum is made by pairs of electronic states, separated by q,
one of which is occupied and the other empty, and both of which have
energies very close to the Fermi level. Consequently, parallel or nesting regions of the Fermi surface tend to produce peaks, known as Kohn
anomalies, at the wave-vector Q which separates them, and it is believed
that the parallel sections of the webbing in the hole surface of Fig. 1.11
give rise to the maxima shown in Fig. 1.17. As we have mentioned, this
conjecture is supported by both positron-annihilation experiments and
band structure calculations but, despite extensive efforts, first-principles
estimates of J (q) have not proved particularly successful. χ(q) may be
calculated quite readily from the energy bands (Liu 1978), and exhibits
the expected peaks, but the exchange matrix elements which determine
I(q) are much less tractable. Lindgård et al. (1975) obtained the correct
general variation with q for Gd, but the matrix elements were, not surprisingly, far too large when the screening of the Coulomb interaction
was neglected.
The Kohn anomalies in J (q) Fourier transform into Friedel oscillations in J (R), and such oscillations, and the extremely long range of the
indirect exchange, are illustrated in the results of Houmann et al. (1979)
for Pr in Fig. 1.18. As is also shown in this figure, they found that the
anisotropic component of the coupling is a substantial proportion of the
Heisenberg exchange. The anisotropic coupling between the moments
on two ions can be written in the general form
HJJ = −
1 Kllmm
(ij)Olm (Ji )Olm (Jj ),
2
(1.4.24)
ij ll mm
where the terms which appear in the sum are restricted by symmetry,
but otherwise may exhibit a large variety, depending on their origin. The
many possible causes of anisotropy have been summarized by Jensen et
al. (1975). They are usually associated with the orbital component of
the moment and are therefore expected to be relatively large when L
is large. In addition to contributions due to the influence of the localized 4f orbital moment on the conduction electrons (Kaplan and Lyons
1963), and to the magnetization and spin–orbit coupling of the latter
(Levy 1969), direct multipolar interactions and two-ion magnetoelas
depend explicitly on the
tic couplings, for which the coefficients Kllmm
strain, may be important. A general two-ion coupling which depends
only on the dipolar moments of the 4f electrons is
Hdd = −
1
Jαβ (ij)Jiα Jjβ .
2
ij
(1.4.25)
1.4 MAGNETIC INTERACTIONS
49
Fig. 1.18.
The indirect-exchange interaction between ions on the
hexagonal sites in Pr, deduced from measurements of the magnetic excitations at 6 K. The circles represent the isotropic interaction J (R) between an ion at the origin and those at different sites. The filled symbols are for pairs of ions in the same hexagonal plane, and the open
symbols for pairs in different planes. The former are reasonably well
described by the simple free-electron model of Section 5.7.1, with an effective value of 1.1 Å−1 for 2kF , as shown by the full curve. In addition,
the exchange incorporates an anisotropic component K(R), discussed in
Section 2.1.6, which is smaller, but of comparable magnitude. Its values
between pairs of ions in the plane are indicated by the squares. The calculated uncertainties in the exchange interactions are, at the most, the size
of the points.
The dispersion relations for the magnetic excitations provide extensive
evidence for anisotropy of this form. A special case is the classical dipole–
dipole interaction for which
3(Riα − Rjα )(Riβ − Rjβ ) − δαβ |Ri − Rj |2
.
|Ri − Rj |5
(1.4.26)
Although it is very weak, being typically one or two orders of magnitude
less than the exchange between nearest neighbours, the dipole–dipole
coupling is both highly anisotropic and extremely long-ranged, and may
therefore have important effects on both magnetic structures and excitations. Apart from this example, the anisotropic two-ion couplings are
even more difficult to calculate than are the isotropic components, so
the strategy which has generally been adopted to investigate them is to
Jαβ (ij) = (gµB )2
50
1. ELEMENTS OF RARE EARTH MAGNETISM
assume that all terms in (1.4.24) which are not forbidden by symmetry
are present, to calculate their influence on the magnetic properties, and
to determine their magnitude by judicious experiments.
The hyperfine interaction between the 4f moment and the nuclear
spin I may be written
Hhf = A
Ii · Ji .
(1.4.27)
i
Since A is typically of the order of micro-electron-volts, the coupling to
the nuclei normally has a negligible effect on the electronic magnetism
in the rare earth metals, but we shall see in Sections 7.3 and 7.4 that it
has a decisive influence on the low-temperature ordering in Pr.